Some Notes on the FFT
To do a 2k FFT mod a prime p you need to choose a prime p whose remainders include 2kth roots of unity, and you need to find one such root that is not a 2k-1-th root of unity,
This can be done by taking the 2k-1-st powers mod p of the first few integers until you find
one such that this power is p-1. Get this power by successive squaring, and to be safe mod
on each squaring.
You then want to set up your spreadsheet without dollar signs, so you can copy it freely
We describe a possible set up for this, for k=3 here, so that there are 8 input coefficients of
our polynomial which we enter in opposite to the normal order. Thus to describe the
number 12345678 which is written say, as the polynomial as 78+56*100+ 34*1002
+12*1003 with coefficients in the order 78 56 34 12 0 0 0 0.
I would do this rather inefficiently in space, by putting three rows of the modulus on the
top, with three rows that give the appropriate powers of the 8th root of unity where they
belong.
Then underneath that I would put the input and then put the three rows that do the FFT.
The first row has to be entered in your favorite way. The rest of the rows can be obtained
entirely by copying the first row judiciously/
How?
Suppose you have the first row of the FFT let your data be denoted as
d0
d1
d2
d3
d4
d5
d6
d7
then the first row has the following entries (leaving out the mods and with x your 16th root
of unity)
d0+d4 d0-d4 d1+d5 (d1-d5)*x d2+d6 (d2-d6)*x2 d3+d7 (d3-d7)*x3
(where you have multiplier rows above that look like
1 1 1 x 1 x2 1 x3
1 1 1 1 1 x2 1 x2
1 1 1 1 1 1 1 1
and multiply the appropriate combinations of the row above by the multiplier in these rows
above.
Then to form the second row, you copy the first and fourth and fifth and 8th entrees here
down, and copy the first and fifth to the right into the second and sixth places and the
fourth and eighth to the left into the third and seventh places.
To form the third row you take the first and eighth and copy them down, and then copy the
first to the right into the next three places, and the eighth to the left into the next three
spaces.
A spreadsheet where this is done explicitly with formulae follows.
Here is the spreadsheet:
modulus
8th root of
1
inv of 8
17
2
15
modulus
8th root of
1
inv of 8
17
2
15
*
power
modulus
modulus
0
17
17
1
17
17
2
17
17
3
17
17
4
17
17
5
17
17
6
17
17
7
17
17
modulus
multiplier
multiplier
multiplier
data
data modm
1st round
2nd round
3rd
round=answer
17
1
1
1
78
10
10
4
17
1
1
1
56
5
10
3
17
1
1
1
11
11
5
16
17
2
1
1
12
12
10
0
17
1
1
1
0
0
11
0
17
4
1
1
0
0
10
4
17
1
4
1
0
0
12
6
17
8
4
1
0
0
11
13
4
7
5
13
4
16
10
4
to invert FFT
0
1
17
17
17
17
2
17
17
3
17
17
4
17
17
5
17
17
6
17
17
7
17
17
17
1
1
1
4
4
8
6
17
1
1
1
7
7
0
14
17
1
1
1
5
5
6
10
17
2
1
1
13
13
16
3
17
1
1
1
4
4
15
6
17
4
1
1
16
16
14
3
17
1
4
1
10
10
0
7
17
8
4
1
4
4
4
14
12
10
15
0
0
15
0
0
15
0
0
15
0
0
15
11
12
15
3
11
15
6
5
15
power
modulus
modulus
modulus
multiplier
multiplier
multiplier
data
data modm
1st round
2nd round
3rd
round=answer
div by 8
inv of 8
Notice result on next to last line is input data backwards (last entry is a copy of the 1st
Here is what it looks like with formulae: (showingleft side first:
modulus
8th root of 1
inv of 8
A
17
2
15
B
10
power
modulus
modulus
modulus
multiplier
multiplier
multiplier
data
data modm
1st round
2nd round
3rd
round=answer
0
=B3
=D2
=D3
1
1
1
78
=MOD(D8,D4)
=MOD(D9+H9,D2)
=MOD(D10+H10,D3)
=D1+1
=D2
=D3
=D4
1
1
1
56
=MOD(E8,E4)
=MOD((D9-H9)*E5,E2)
=MOD(E10+I10,E3)
=E1+1
=E2
=E3
=E4
1
1
1
11
=MOD(F8,F4)
=MOD(E9+I9,E2)
=MOD((D10-H10)*F6,F3)
=MOD(D11+H11,D4)
=MOD(E11+I11,E4)
=MOD(F11+J11,F4)
C
D
to invert FFT
E
F
modulus
8th root of 1
inv of 8
17
2
15
:
power
modulus
modulus
modulus
multiplier
multiplier
multiplier
data
data modm
0
=B18
=D17
=D18
1
1
1
=D12
=MOD(D23,D19)
1st round
=MOD(D24+H24,D17)
=D16+1
=D17
=D18
=D19
1
1
1
=E12
=MOD(E23,E19)
=MOD((D24H24)*E20,E17)
=E16+1
=E17
=E18
=E19
1
1
1
=F12
=MOD(F23,F19)
2nd round
3rd
round=answer
div by 8
inv of 8
=MOD(D25+H25,D18)
=MOD(E25+I25,E18)
=MOD(E24+I24,E17)
=MOD((D25H25)*F21,F18)
=MOD(D26+H26,D19)
=MOD(D27*D29,D17)
=B20
=MOD(E26+I26,E19)
=MOD(E27*E29,E17)
=D29
=MOD(F26+J26,F19)
=MOD(F27*F29,F17)
=E29
=F1+1
=G1+1
=H1+1
=I1+1
=J1+1
1
=F2
=G2
=H2
=I2
=J2
2
=F3
=G3
=H3
=I3
=J3
3
=F4
=G4
=H4
=I4
=J4
4
=B4
1
=MOD(G5*G5,I2)
1
=MOD(I5*G5,K2)
5
1
1
1
=I5
=J6
6
1
1
1
1
1
7
12
0
0
0
0
8
=MOD(G8,G4)
=MOD(H8,H4)
=MOD(I8,I4)
=MOD(J8,J4)
=MOD(K8,K4)
9
=MOD((E9-I9)*G5,G2)
=MOD(F9+J9,F2)
=MOD((F9-J9)*I5,I2)
=MOD(G9+K9,G2)
1
=MOD((E10-I10)*G6,G3)
=MOD(F10+J10,F3)
=MOD(G10+K10,G3)
=MOD((F10-J10)*J6,J3)
=MOD(G11+K11,G4)
=MOD((D11-H11)*H7,H4)
=MOD((E11-I11)*I7,I4)
=MOD((F11-J11)*J7,J4)
=MOD((G9-K9)*K5,K2)
=MOD((G10K10)*K6,K3)
=MOD((G11K11)*K7,K4)
1
1
1
1
1
=F16+1
=G16+1
=H16+1
=I16+1
=J16+1
1
=F17
=G17
=H17
=I17
=J17
1
=F18
=G18
=H18
=I18
=J18
1
=F19
=G19
=H19
=I19
=J19
1
=B19
1
=MOD(G20*G20,I17)
1
=MOD(I20*G20,K17)
2
1
1
1
=I20
=J21
2
1
1
1
1
1
2
=G12
=H12
=I12
=J12
=K12
2
=MOD(G23,G19)
=MOD((E24I24)*G20,G17)
=MOD((E25I25)*G21,G18)
=MOD(H23,H19)
=MOD(I23,I19)
=MOD((F24J24)*I20,I17)
=MOD(J23,J19)
2
=MOD(G25+K25,G18)
=MOD((F25-J25)*J21,J18)
=MOD(G26+K26,G19)
=MOD(F25+J25,F18)
=MOD((D26H26)*H22,H19)
=MOD((E26-I26)*I22,I19)
=MOD((F26-J26)*J22,J19)
=MOD(K23,K19)
=MOD((G24K24)*K20,K17)
=MOD((G25K25)*K21,K18)
=MOD((G26K26)*K22,K19)
=MOD(G27*G29,G17)
=MOD(H27*H29,H17)
=MOD(I27*I29,I17)
=MOD(J27*J29,J17)
=MOD(K27*K29,K17)
=
=F29
=G29
=H29
=I29
=J29
2
=MOD(F24+J24,F17)
=MOD(G24+K24,G17)
2
2
2